Expert Solutions | Senioritis

John Rhodes
August 16, 2023
Computational Mathematics

Divisible Palindromes

The numbers $545$, $5\,995$ and $15\,151$ are the three smallest palindromes divisible by $109$. There are nine palindromes less than $100\,000$ which are divisible by $109$. How...

John Rhodes
August 16, 2023
Computational Mathematics

Neighbourly Constraints

Let $T(n, m)$ be the number of $m$-tuples of positive integers such that the sum of any two neighbouring elements of the tuple is $\le n$. For...

John Rhodes
August 16, 2023
Computational Mathematics

Frictionless Tube

Consider a horizontal frictionless tube with length $L$ millimetres, and a diameter of 20 millimetres. The east end of the tube is open, while the west end...

John Rhodes
August 16, 2023
Computational Mathematics

Distinct Values of a Proto-logarithmic Function

Consider the values of $\log_2(8)$, $\log_4(64)$ and $\log_3(27)$. All three are equal to $3$. Generally, the function $f(m,n)=\log_m(n)$ over integers $m,n \ge 2$ has the property that...

John Rhodes
August 16, 2023
Computational Mathematics

An infinitely long cylinder has its curved surface fully covered with different coloured but otherwise identical rectangular stickers, without overlapping. The stickers are aligned with the cylinder,...

John Rhodes
August 16, 2023
Computational Mathematics

Divisors of Binomial Product

Let $B(n) = \displaystyle \prod_{k=0}^n {n \choose k}$, a product of binomial coefficients. For example, $B(5) = {5 \choose 0} \times {5 \choose 1} \times {5 \choose...

John Rhodes
August 16, 2023
Computational Mathematics

Low-Prime Chessboard Nim

Alice and Bob are taking turns playing a game consisting of $c$ different coins on a chessboard of size $n$ by $n$. The game may start with...

John Rhodes
August 16, 2023
Computational Mathematics

Skipping Squares

For some fixed $\rho \in [0, 1]$, we begin a sum $s$ at $0$ and repeatedly apply a process: With probability $\rho$, we add $1$ to $s$,...